MHEC 51023: Basic Statistics | Handout #1
Learning Objectives
After reading this handout, you will be able to:
- Understand basic probability concepts and terminology
- Calculate probabilities using set operations
- Apply conditional probability in health settings
- Use Bayes’ theorem for diagnostic test interpretation
- Connect probability theory to health economics decisions
1. Introduction: Why Probability Matters in Health Economics
Health economics is full of uncertainty. We do not know who will get sick. We do not know which treatment will work best. We do not know exact costs. Probability helps us measure this uncertainty with numbers.
Probability measures how likely an event is to happen. It is a number between 0 and 1. Zero means impossible. One means certain.
Statistics gives us methods to collect data, analyse it, and make conclusions. We use statistics to make decisions when we do not have complete information.
Statistics has two branches:
- Descriptive statistics: This summarises data. It includes tables, graphs, averages, and measures of spread.
- Inferential statistics: This uses samples to make statements about larger populations. It includes estimation, confidence intervals, and hypothesis tests.
In health economics, we often work with samples. For example, a clinical trial may test a drug on 1,000 patients. But we want to know if the drug works for millions of people. Inferential statistics helps us do this. Probability is the foundation for these methods (Briggs, Claxton and Sculpher, 2006).
2. Basic Definitions
2.1 What is Probability?
When many outcomes are possible, probability tells us the chance of each outcome. If all outcomes are equally likely, we can use a simple formula:
For example, if you roll a fair die, each number (1 to 6) has the same chance. The probability of rolling a 3 is 1 ÷ 6.
2.2 Sample Space (Ω)
The sample space is the set of all possible outcomes. We use the Greek letter Ω (omega) for it.
Sample spaces can be:
- Finite: Limited number of outcomes (e.g., coin toss: {Heads, Tails})
- Countably infinite: Outcomes can be listed forever (e.g., number of hospital visits)
- Uncountably infinite: Too many to count (e.g., exact waiting time at a clinic)
• Blood type test: Ω = {A, B, AB, O}
• Patient outcome after surgery: Ω = {Full recovery, Partial recovery, No improvement, Death}
• Number of patients arriving at ER in one hour: Ω = {0, 1, 2, 3, …} (countably infinite)
2.3 Events
An event is any subset of the sample space. It is a collection of outcomes we are interested in.
Sample space: Blood pressure readings = {Normal, Elevated, High Stage 1, High Stage 2, Crisis}
Event A: “Patient has high blood pressure” = {High Stage 1, High Stage 2, Crisis}
Event B: “Patient needs immediate treatment” = {Crisis}
2.4 Special Events
Two special events are important:
- Null event (∅): The impossible event. It has no outcomes. P(∅) = 0.
- Certain event (Ω): The event that always happens. It contains all outcomes. P(Ω) = 1.
If a hospital only treats adults (age 18+), the event “patient is under 18” is a null event for that hospital’s patient database. The probability is zero.
3. Axioms of Probability
The mathematician Andrey Kolmogorov created three rules that all probabilities must follow. These are called the Kolmogorov axioms (Kolmogorov, 1933):
Axiom 1: Every probability is between 0 and 1. (0 ≤ P(A) ≤ 1)
Axiom 2: The probability of nothing happening is zero. P(∅) = 0
Axiom 3: The probability of something happening is one. P(Ω) = 1
From these axioms, we get the addition rule:
We subtract P(A and B) because we do not want to count outcomes twice.
Think of it like counting people. If 30 people have diabetes and 25 people have hypertension, you cannot simply add 30 + 25 = 55. Some people have both conditions. You must subtract those people to avoid counting them twice.
4. Set Operations on Events
We use set operations to combine events. These operations help us calculate complex probabilities.
4.1 Complement (Aᶜ)
The complement of A is everything NOT in A. If A happens, Aᶜ does not happen.
A medical device factory tests products. The probability of a device having no more than 2 defects is 0.86.
Question: What is the probability of more than 2 defects?
Solution: P(more than 2 defects) = 1 − 0.86 = 0.14 or 14%
4.2 Intersection (A ∩ B)
The intersection is when both A AND B happen together. This is called a joint event.
In a hospital study, A = “patient is female” and B = “patient has diabetes”. The intersection A ∩ B = “female patient with diabetes”. This joint probability is important for understanding disease patterns in different groups (Rothman, Greenland and Lash, 2008).
4.3 Mutually Exclusive Events
Two events are mutually exclusive (or disjoint) if they cannot happen at the same time. Their intersection is empty.
A patient’s blood type can only be one type. The events “blood type A” and “blood type B” are mutually exclusive.
If P(Type A) = 0.40 and P(Type B) = 0.11, then:
P(Type A or Type B) = 0.40 + 0.11 = 0.51
Mutually exclusive ≠ Independent. These are different concepts. Mutually exclusive events cannot happen together. Independent events do not affect each other’s probability.
4.4 Union (A ∪ B)
The union is when A OR B happens (or both). At least one of them occurs.
5. Probability Calculations: Health Economics Examples
5.1 Healthcare Workforce Analysis
A health ministry surveys 200 nurses. They record salary range and years of experience.
| Salary Range (USD) | 1-5 Years | 6-10 Years | 11-15 Years | Total |
|---|---|---|---|---|
| 30,000 – 49,999 | 8 | 20 | 40 | 68 |
| 50,000 – 69,999 | 10 | 32 | 30 | 72 |
| 70,000+ | 20 | 30 | 10 | 60 |
| Total | 38 | 82 | 80 | 200 |
Calculating Probabilities from the Table:
Joint probability: Probability of two events happening together.
P(Low salary AND 6-10 years experience) = 20 ÷ 200 = 0.10 or 10%
Marginal probability: Probability of a single event (from the margins).
P(Medium salary) = 72 ÷ 200 = 0.36 or 36%
Union probability: Using the general addition rule.
P(Low salary OR 6-10 years) = (68/200) + (82/200) − (20/200) = 130/200 = 0.65 or 65%
This type of analysis is common in health workforce planning. Policy makers use these probabilities to understand salary distributions and plan budgets (World Health Organization, 2016).
6. Independent Events and the Multiplication Rule
Two events are independent if one happening does not change the probability of the other.
If P(patient is male) = 0.48 and P(has insurance) = 0.75, and these events are independent:
P(Male AND Insured) = 0.48 × 0.75 = 0.36
But be careful! In real health data, gender and insurance status are often NOT independent. This is why we need data to check (Wooldridge, 2010).
6.1 Dependent Events: Sampling Without Replacement
When events are NOT independent, we use the general multiplication rule:
The symbol P(B | A) means “probability of B given that A has happened”.
A batch has 10 vaccine vials. Six are safe, four are contaminated. A technician tests 2 vials without replacement.
Question: What is P(both vials are safe)?
P(1st safe) = 6/10
P(2nd safe | 1st safe) = 5/9 (one safe vial removed)
P(both safe) = (6/10) × (5/9) = 30/90 = 1/3 ≈ 0.33
7. Conditional Probability
Conditional probability is one of the most important concepts in health economics. It tells us the probability of one event given that another event has occurred.
P(A | B) is read as “probability of A given B”. The vertical line | means “given that”.
Conditional probability is central to diagnostic testing:
• P(Disease | Positive test) = probability of having disease given a positive test
• P(Positive test | Disease) = probability of positive test given the person has disease
These two probabilities are NOT the same! This confusion is called the “prosecutor’s fallacy” and leads to serious errors in medical decision-making (Gigerenzer, 2002).
8. Counting Rules
Sometimes we need to count how many outcomes are possible. The product rule helps with this.
A clinical trial tests:
• 3 different drug doses
• 2 delivery methods (pill vs injection)
• 4 treatment durations
Total combinations = 3 × 2 × 4 = 24 combinations
This helps researchers plan resources and understand the complexity of factorial designs (Friedman, Furberg and DeMets, 2015).
9. Bayes’ Theorem: The Most Important Tool in Health Economics
Bayes’ theorem lets us update probabilities when we get new information. It was developed by Thomas Bayes in the 18th century but is now essential in health economics.
In words:
- P(H) = Prior probability (what we believed before evidence)
- P(E | H) = Likelihood (probability of evidence if hypothesis is true)
- P(H | E) = Posterior probability (updated belief after evidence)
9.1 Extended Form for Multiple Hypotheses
When there are several possible explanations (B₁, B₂, … Bₖ), we use:
The bottom part is called the total probability. It sums up all ways that A can happen.
10. Applications of Bayes’ Theorem
10.1 Diagnostic Testing
This is perhaps the most important application in health economics.
Given information:
• Disease prevalence: P(D) = 1% (0.01)
• Test sensitivity: P(T⁺ | D) = 99%
• Test specificity: P(T⁻ | Dᶜ) = 95%
Question: If a patient tests positive, what is the probability they have the disease?
Step-by-step Solution:
Step 1: We want P(D | T⁺) — probability of disease given positive test.
Step 2: Calculate false positive rate.
P(T⁺ | Dᶜ) = 1 − 0.95 = 0.05
Step 3: Calculate P(T⁺) using total probability.
P(T⁺) = P(T⁺ | D) × P(D) + P(T⁺ | Dᶜ) × P(Dᶜ)
P(T⁺) = (0.99 × 0.01) + (0.05 × 0.99) = 0.0099 + 0.0495 = 0.0594
Step 4: Apply Bayes’ theorem.
P(D | T⁺) = (0.99 × 0.01) ÷ 0.0594 = 0.0099 ÷ 0.0594 ≈ 0.167 or 16.7%
This result surprises many people. Even with a “99% accurate” test, a positive result only means a 16.7% chance of disease! This is because the disease is rare. Most positive tests are false positives from healthy people. This has huge implications for screening policy and cost-effectiveness (Moons et al., 2015).
10.2 Supply Chain Quality
Three factories (A, B, C) supply hospital lamps:
• Factory A supplies 35% of lamps, 2% defect rate
• Factory B supplies 40% of lamps, 3% defect rate
• Factory C supplies 25% of lamps, 5% defect rate
Question: A lamp is found defective. What is the probability it came from Factory C?
Solution:
P(Defective) = (0.02 × 0.35) + (0.03 × 0.40) + (0.05 × 0.25)
= 0.007 + 0.012 + 0.0125 = 0.0315
P(C | Defective) = (0.05 × 0.25) ÷ 0.0315 = 0.0125 ÷ 0.0315 ≈ 0.397 or 39.7%
This analysis helps hospitals decide which suppliers to audit or replace. Even though Factory C only supplies 25% of lamps, it is responsible for almost 40% of defects. This type of analysis is part of health technology assessment (Drummond et al., 2015).
10.3 Differential Diagnosis
During an outbreak, patients come with fever. Some have flu, some have measles.
• P(Flu) = 0.70 (70% of fever cases are flu)
• P(Measles) = 0.30
• P(Rash | Flu) = 0.08 (8% of flu patients get rash)
• P(Rash | Measles) = 0.95 (95% of measles patients get rash)
Question: A patient has a rash. What is P(Measles | Rash)?
Solution:
P(Rash) = (0.08 × 0.70) + (0.95 × 0.30) = 0.056 + 0.285 = 0.341
P(Measles | Rash) = (0.95 × 0.30) ÷ 0.341 = 0.285 ÷ 0.341 ≈ 0.836 or 83.6%
Although measles is less common overall, a rash strongly suggests measles because rashes are rare in flu.
Key Takeaways
1. Probability basics: Probability measures uncertainty on a 0-1 scale. Sample spaces contain all outcomes. Events are subsets we care about.
2. Set operations: Use complement, intersection, and union to combine events. The addition rule helps calculate P(A or B).
3. Independence vs dependence: Independent events use P(A and B) = P(A) × P(B). Dependent events require conditional probability.
4. Conditional probability: P(A | B) tells us the probability of A given B has occurred. This is central to diagnostic testing.
5. Bayes’ theorem: Updates probabilities with new evidence. Essential for interpreting diagnostic tests and making health policy decisions.
Practice Questions
In a population: 45% have private insurance, 30% have public insurance, 10% have both. What is the probability a person has some form of insurance?
Hint: Use the addition rule.
A drug has a 70% success rate. If 3 independent patients take the drug, what is the probability all 3 are cured?
Hint: Use the multiplication rule for independent events.
A cancer has 2% prevalence. A screening test has 90% sensitivity and 85% specificity. Calculate P(Cancer | Positive test).
Hint: Use Bayes’ theorem.
References
Briggs, A., Claxton, K. and Sculpher, M. (2006) Decision modelling for health economic evaluation. Oxford: Oxford University Press.
Drummond, M.F., Sculpher, M.J., Claxton, K., Stoddart, G.L. and Torrance, G.W. (2015) Methods for the economic evaluation of health care programmes. 4th edn. Oxford: Oxford University Press.
Friedman, L.M., Furberg, C.D. and DeMets, D.L. (2015) Fundamentals of clinical trials. 5th edn. New York: Springer.
Gigerenzer, G. (2002) Calculated risks: how to know when numbers deceive you. New York: Simon & Schuster.
Kolmogorov, A.N. (1933) Grundbegriffe der wahrscheinlichkeitsrechnung. Berlin: Springer.
Moons, K.G.M., Altman, D.G., Reitsma, J.B., Ioannidis, J.P.A., Macaskill, P., Steyerberg, E.W., Vickers, A.J., Ransohoff, D.F. and Collins, G.S. (2015) ‘Transparent reporting of a multivariable prediction model for individual prognosis or diagnosis (TRIPOD): explanation and elaboration’, Annals of Internal Medicine, 162(1), pp. W1-W73.
Rothman, K.J., Greenland, S. and Lash, T.L. (2008) Modern epidemiology. 3rd edn. Philadelphia: Lippincott Williams & Wilkins.
Wooldridge, J.M. (2010) Econometric analysis of cross section and panel data. 2nd edn. Cambridge, MA: MIT Press.
World Health Organization (2016) Global strategy on human resources for health: workforce 2030. Geneva: World Health Organization.
Next in this series: Handout #2 will cover Random Variables and Probability Distributions.
